Eq. REF  shows that at long times, the out-of-diagonal terms give destructive interference and only the diagonal time-independent ones survive. Unless particular symmetries are present in the spectrum this is generally fulfilled, and the thermodynamical average of the property \(O\)  is only given by the initial condition of the system. Interestingly, according to the Eigenstate Thermalization Hypothesis (ETH)[1]}, if this initial condition fulfills some specific conditions, the final average will be consistent with statistical mechanics. This is not always the case and systems exhibiting many-body localization are subject of intense research[2]}. In this work we will always assume that thermalization is possible and we will instead focus on determining the rate at which this thermalized state is reached.
